The p2 0q Superconformal Bootstrap · The p2;0qtheories as abstract CFTs No intrinsic eld-theoretic...

The p2, 0q Superconformal Bootstrap

Leonardo Rastelli

Yang Institute for Theoretical Physics

Stony Brook

Based on work withChris Beem, Madalena Lemos and Balt van Rees

Physics on the RivieraSestri Levante, Sept 16 2015

p2, 0q theories

Nahm’s classification: superconformal algebras exist for d ď 6.In d “ 6, pN , 0q algebras. Existence of Tµν multiplet requires N ď 2.

p2, 0q: maximal susy in maximal d. No marginal couplings allowed.

Interacting models inferred from string/M-theory: ADE catalogue.

Central to many recent developments in QFT.“Mothers” of many interesting QFTs in d ă 6.

Key properties:

Moduli space of vacua

Mg “ pR5qrg{Wg, g “ tAn, Dn, E6, E7, E8u.

On R5 ˆ S1, IR description as 5d MSYM with gauge algebra g .

At large n, An and Dn theories described through AdS/CFT:M-theory on AdS7 ˆ S

4 and AdS7 ˆ RP4.

Leonardo Rastelli (YITP) p2, 0q Bootstrap Sept ’15 1 / 22

The p2, 0q theories as abstract CFTs

No intrinsic field-theoretic formulation yet.No conventional Lagrangian (hard to imagine one from RG lore).

Working hypothesis: (at least) for correlators of local operators in R6,the p2, 0q theory is just another CFT, defined by a local operator algebra

OPE : O1pxqO2p0q “ÿ

k

c12kpxqOkp0q

Can symmetry and basic consistency requirements completely determinethe spectrum and OPE coefficients?


Abstract CFT Framework

A general Conformal Field Theory hasn’t much to do with “fields”(of the kind we write in Lagrangians).

We’ll think more abstractly. A CFT is defined by its local operators,

A ” tOkpxqu ,

and their correlation functions xO1px1q . . .Onpxnqy .

A is an algebra. Operator Product Expansion (OPE),

O1pxqO2p0q “ÿ

k

c12k pOkp0q ` . . . q ,

where the . . . are fixed by conformal invariance. The sum converges.

Caveat I: This definition does not capture non-local observables, such as

conformal defects. (E.g., Wilson lines in a conformal gauge theory.).


Reduce npt to pn´ 1qpt,

xO1px1qO2px2q . . .Onpxnqy “ÿ

k

c12kpx2q xOkpx2q . . .Onpxnqy .

1pt functions are trivial, xOipxqy “ 0 except for x1y ” 1.

O∆,`,f pxq labeled by conformal dimension ∆, Lorentz representation ` andpossibly flavor quantum number f .

The CFT data tp∆i, `i, fiq , cijku completely specify the theory.

But not anything goes! Consistency conditions:

Associativity:pO1O2qO3 “ O1 pO2O3q .

Unitarity (reflection positivity):Lower bounds on ∆ for given `;cijk P R

Caveat II: In non-trivial geometries, xOy ‰ 0 Ñ additional constraints.

In d “ 2, modularity. In d ą 2, harder to analyze, have been ignored so far.Leonardo Rastelli (YITP) p2, 0q Bootstrap Sept ’15 4 / 22

The bootstrap program

Old aspiration (1970s) Ferrara Gatto Grillo, Polyakov.Associativity ” crossing symmetry of 4pt functions

=!

O!

!

O OO!

1

1

2

2 4

4

3

3

1

Vastly over-constrained system of equations for t∆i, cijku.Classification and construction of CFTs reduced to an algebraic problem.

‚ Famous success story in d “ 2, starting from BPZZ (1984).2d conformal symmetry is infinite dimensional, z Ñ fpzq.In some cases, finite-dimensional bootstrap problem (rational CFTs).Many exact solutions, partial classification.


Bootstrapping in two steps

For d “ 6, N “ p2, 0q SCFTs (as well as d “ 4, N ě 2 SCFTs)the crossing equations split into

(1) Equations that depend only on theintermediate BPS operators. Captured by the 2d chiral algebra.“Minibootstrap”

(2) Equations that also includeintermediate non-BPS operators.“Maxibootstrap”

(1) are tractable and determine an infinite amount of CFT data.

This is essential input to the full-fledged bootstrap (2),which can be studied numerically.

Beem Lemos Liendo Peelaers LR van Rees, Beem LR van Rees


Meromorphy in p2, 0q SCFTs

Fix a plane R2 Ă R6, parametrized by pz, zq.

Claim : D subsector Aχ “ tOipzi, ziqu with meromorphic

xO1pz1, z1qO2pz2, z2q . . .Onpzn, znqy “ fpziq .

Rationale: Aχ ” cohomology of a nilpotent Q ,

Q “ Q` S ,

Q Poincare, S conformal supercharges.

z dependence is Q -exact: cohomology classes rOpz, zqsQ Opzq.Analogous to the d “ 4, N “ 1 chiral ring:cohomology classes rOpxqsQ 9α

are x-independent.


χ6 : 6d (2,0) SCFT ÝÑ 2d Chiral Algebra.

Global slp2q Ñ Virasoro, indeed T pzq :“ rΦpIJqpz, zqsQ ,with ΦpIJq the stress-tensor multiplet superprimary.

c2d “ c6d

in normalizations where c6d (free tensor) ” 1.

All 12 -BPS operators p∆ “ 2R) are in Q cohomology.

Generators of the 12 -BPS ring Ñ generators of the chiral algebra.

Some semi-short multiplets also play a role.


Chiral algebra for p2, 0q theory of type AN´1

One 12 -BPS generator each of dimension ∆ “ 4, 6, . . . 2N

ó

One chiral algebra generator each of dimension h “ 2, 3, . . . N.

Most economical scenario: these are all the generators.Check: the superconformal index computed by Kim3 is reproduced.

Plausibly a unique solution to crossing for this set of generators.

The chiral algebra of the AN´1 theory is WN , with

c2d “ 4N3 ´ 3N ´ 1 .

(Similar proposals for D and E theories).


Half-BPS 3pt functions of p2, 0q SCFT

OPE of Wg generators ñ half-BPS 3pt functions of SCFT.

Let us check the result at large N .

WNÑ8 with c2d „ 4N3 Ñ a classical Poisson algebra.We can use results on universal Poisson algebra W8rµs, with µ “ N .(Gaberdiel Hartman, Campoleoni Fredenhagen Pfenninger)

We find

Cpk1, k2, k3q “22α´2

pπNq32

Γ´α

2

¯

˜

Γ`

k123`12

˘

Γ`

k231`12

˘

Γ`

k312`12

˘

a

Γp2k1 ´ 1qΓp2k2 ´ 1qΓp2k3 ´ 1q

¸

kijk ” ki ` kj ´ kk, α ” k1 ` k2 ` k3,in precise agreement with calculation in 11d sugra on AdS7 ˆ S

4!(Corrado Florea McNees, Bastianelli Zucchini)

1{N corrections in WN OPE ñ quantum M-theory corrections.


p2, 0q maxibootstrap Beem Lemos LR van Rees

Universal 4pt function of ΦpIJq, superprimary of Tµν multiplet.

Unique structure in superspace.

Only input: 6d Weyl anomaly coefficient c.

For ADE theories,c “ 4dgh

_g ` rg ,

but we keep it general.


Double OPE expansion

xΦΦΦΦy “ÿ

OPΦˆΦ

f2ΦΦOG

ΦO

We impose the absence of higher-spin currents.

The Os P Φˆ Φ are:

Infinite set tOχu of Q -chiral BPS multiplets, fixed from χ-algebra.

Infinite tower of BPS multiplet tD,B1,B3, . . . u, not in χ-algebra.

Infinite set of non-BPS multiplets L∆,`, sop5qR singlets.Bose symmetry Ñ ` is even. Unitarity bound ∆ ě `` 6.

Unfixed BPS multiplets correspond to long multiplets at threshold,

lim∆Ñ``6

GΦL∆,`

“ GΦB`´1

pD ” B´1q ,


Bootstrap sum rule

=!

O!

!

O OO!

1

1

2

2 4

4

3

3

1

When the dust settles, a single sum ruleÿ

long superprimaries

f2∆,` F∆,`pz, zq ` Fχpz, z; cq “ 0

z, z: conformal cross ratios;

F∆,` ” G∆,` ´ Gˆ∆,`: superconformal block minus its crossing;

Fχpz, z; cq: an explicitly known function (from minibootstrap).

The unknown CFT data to be constrained are:

Set of (dimension, spin) tp∆i, `iqu of the intermediate multiplets.

The (squared) OPE coefficients f2∆i,`i

. Non-negative by unitarity.


The numerical oracle (Rattazzi Rychkov Tonni Vichi)

ÿ

p∆i,`iq

f2∆,` F∆,`pz, zq ` Fknownpz, z; cq “ 0

Use the sum rule to constrain the space of CFT data.

For example, consider a trial spectrum with ∆ ě ∆` for operators of spin `.If there exists a linear functional χ such that

χ ¨ F∆,`pz, zq ě 0 when ∆ ě ∆`

χ ¨ Fknownpz, z; cq “ 1

that trial spectrum is ruled out – oracle says NO.If one cannot find such a χ, oracle says MAYBE.

Implemented by linear programming or semi-definite programming.Surprisingly powerful!


Scalar bound in general d “ 3 CFT[El-Showk, Paulos, Poland, Rychkov, Simmons-Duffin and Vichi, PRD 86, 025022]

Exclusion plot in the subspace of d “ 3 CFT data p∆σ,∆εq withσ ˆ σ “ 1` ε` . . . , from the bootstrap of a single 4pt function xσσσσy.

Two real surprises:

3d Ising appears to lie on the exclusion curve(i.e. it saturates the bound)

3d Ising appears to sit at a special kink on the exclusion curve.


Multiple Correlators [Kos, Poland, Simmons-Duffin, ‘14]

CFT3 with Z2 symmetry. σ odd, ε even, σ ˆ σ “ 1` ε` . . .System of correlators xσσσσy, xσσεεy, xεεεεy.Allowed region assuming that only one odd scalar is relevant p∆σ1 ě 3q:

allowed region with !!! ! 3 (nmax = 6)

!!

!"

0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.581

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Figure 2: Allowed region of (!!,!") in a Z2-symmetric CFT3 where !!! ! 3 (only oneZ2-odd scalar is relevant). This bound uses crossing symmetry and unitarity for "!!!!#,"!!""#, and """""#, with nmax = 6 (105-dimensional functional), #max = 8. The 3D Ising pointis indicated with black crosshairs. The gap in the Z2-odd sector is responsible for creating asmall closed region around the Ising point.

The allowed region around the Ising point shrinks further when we increase the valueof nmax. Finding the allowed region at nmax = 10 (N = 275) is computationally intensive,so we tested only the grid of 700 points shown in figure 5. The disallowed points in thefigure were excluded by assuming both !!! ! 3 and !"! ! 3. On the same plot, we alsoshow the nmax = 14 single-correlator bound on !" computed in [22] using a very di"erentoptimization algorithm. The final allowed region is the intersection of the region below thenmax = 14 curve and the region indicated by our allowed multiple correlator points.

Since the point corresponding to the 3D Ising model must lie somewhere in the allowedregion, we can think of the allowed region as a rigorous prediction of the Ising modeldimensions, giving !! = 1/2 + !/2 = 0.51820(14) and !" = 3 $ 1/" = 1.4127(11). Infigure 6 we compare our rigorous bound with the best-to-date predictions using MonteCarlo simulations [35] and the c-minimization conjecture [22]. Although our result has un-certainties greater than c-minimization by a factor of %10 and Monte-Carlo determinationsby a factor of %3, they still determine !! and !" with 0.03% and 0.08% relative uncertainty,respectively. Increasing nmax further could potentially lead to even better determinations of!! and !". Indeed, the single correlator bound at nmax = 14 passing through the allowedregion in figure 5 indicates that the nmax = 10 allowed region is not yet optimal. At thispoint, it is not even clear whether continually increasing nmax might lead to a finite allowed

25

3d Ising gets cornered! ∆σ “ 0.518151p5q, ∆ε “ 1.41263p5q,most accurate to date [Simmons-Duffin ’15]


A lower bound on c

There is a minimum anomaly cmin compatible with crossing and unitarity.The bound cmin increases as we increase the search space for thefunctional, parametrized by a cutoff Λ.

Extrapolating, cmin Ñ 25, the value of the A1 theory (” two M5s)!

(We are disallowing the free theory pc “ 1q by forbidding HS currents.)


For c ă cmin, the oracle says NO. Why?

For c ă cmin, solutions to crossing have λ2D ă 0, violating unitarity.

λ2D “ 0 precisely at c “ cmin.

Agrees with conjecture of Batthacharyya and Minwalla about 14BPS partition

function of A1 theory: D multiplet absent!


Bootstrapping the A1 theory

For c “ cmin Ñ 25, D unique unitary solution to crossing.

Claim: The A1 theory can be completely bootstrapped!

Upper bounds on the dimension of the leading-twist unprotected scalar,under different assumptions. Perfectly consistent.


Exclusion region in p∆0,∆2q plane for c “ 25 (A1 value).

The corner values are conjectured to be the true leading-twist dimensionsof the physical A1 theory.


General c

2 5 10 206.0

6.5

7.0

7.5

8.0

8.5

9.0

c1�3

D0

22 derivatives

21 derivatives

20 derivatives

19 derivatives

18 derivatives

17 derivatives

16 derivatives

15 derivatives

14 derivatives

A1 theory

2 5 10 208.0

8.5

9.0

9.5

10.0

10.5

11.0

c1�3

D2

22 derivatives

21 derivatives

20 derivatives

19 derivatives

18 derivatives

17 derivatives

16 derivatives

15 derivatives

14 derivatives

A1 theory

Bounds for the leading-twist unprotected operators of spin ` “ 0, 2.

For cÑ8, they appear to be saturated by AdS7 ˆ S4 sugra,

including 1{c corrections.

For large c, leading-twist unprotected operators are double-traces of theform Os “ O14B

sO14, with ∆s “ 8` s´Op1{cq.

Summary: Both at small and large c the bootstrap bounds appear to besaturated by physical p2, 0q theories.


Outlook

The p2, 0q theories can be successfully studied by bootstrap methods.

Exact results from the chiral algebra, e.g. 12 BPS 3pt functions.

Systematic 1{N expansion and its M-theory interpretation?

A derivation of the AGT correspondence?Codimension-two defects ñ Toda vertex operators?

Numerical results for the non-protected spectrum.

A1 theory in principle completely cornered by bootstrap equations.Beginning of a systematic algorithm to solve it.

Aną1 theories need input on BPS spectrum and multiple correlators.

Precision numerics? Multiple correlators?

Further analytic insights?


The p2 0q Superconformal Bootstrap · The p2;0qtheories as abstract CFTs No intrinsic eld-theoretic...

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